Is ${980422}$ divisible by $9$ ?
Answer: A number is divisible by $9$ if the sum of its digits is divisible by $9$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {980422}= &&{9}\cdot100000+ \\&&{8}\cdot10000+ \\&&{0}\cdot1000+ \\&&{4}\cdot100+ \\&&{2}\cdot10+ \\&&{2}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {980422}= &&{9}(99999+1)+ \\&&{8}(9999+1)+ \\&&{0}(999+1)+ \\&&{4}(99+1)+ \\&&{2}(9+1)+ \\&&{2} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {980422}= &&\gray{9\cdot99999}+ \\&&\gray{8\cdot9999}+ \\&&\gray{0\cdot999}+ \\&&\gray{4\cdot99}+ \\&&\gray{2\cdot9}+ \\&& {9}+{8}+{0}+{4}+{2}+{2} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $9$ , so the first five terms must all be multiples of $9$ That means that to figure out whether the original number is divisible by $9 $ , all we need to do is add up the digits and see if the sum is divisible by $9$ . In other words, ${980422}$ is divisible by $9$ if ${ 9}+{8}+{0}+{4}+{2}+{2}$ is divisible by $9$ Add the digits of ${980422}$ $ {9}+{8}+{0}+{4}+{2}+{2} = {25} $ If ${25}$ is divisible by $9$ , then ${980422}$ must also be divisible by $9$ ${25}$ is not divisible by $9$, therefore ${980422}$ must not be divisible by $9$.